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Register a new user$omega^omega$-Base and infinite-dimensional compact sets in locally convex spaces
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Johannes Philipp Sch\\\"urz
Publication date
2020
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English
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A locally convex space (lcs) $E$ is said to have an $omega^{omega}$-base if $E$ has a neighborhood base ${U_{alpha}:alphainomega^omega}$ at zero such that $U_{beta}subseteq U_{alpha}$ for all $alphaleqbeta$. The class of lcs with an $omega^{omega}$-base is large, among others contains all $(LM)$-spaces (hence $(LF)$-spaces), strong duals of distinguished Frechet lcs (hence spaces of distributions $D(Omega)$). A remarkable result of Cascales-Orihuela states that every compact set in a lcs with an $omega^{omega}$-base is metrizable. Our main result shows that every uncountable-dimensional lcs with an $omega^{omega}$-base contains an infinite-dimensional metrizable compact subset. On the other hand, the countable-dimensional space $varphi$ endowed with the finest locally convex topology has an $omega^omega$-base but contains no infinite-dimensional compact subsets. It turns out that $varphi$ is a unique infinite-dimensional locally convex space which is a $k_{mathbb{R}}$-space containing no infinite-dimensional compact subsets. Applications to spaces $C_{p}(X)$ are provided.
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A topological space $X$ is defined to have an $omega^omega$-base if at each point $xin X$ the space $X$ has a neighborhood base $(U_alpha[x])_{alphainomega^omega}$ such that $U_beta[x]subset U_alpha[x]$ for all $alphalebeta$ in $omega^omega$. We characterize topological and uniform spaces whose free (locally convex) topological vector spaces or free (Abelian or Boolean) topological groups have $omega^omega$-bases.
The concept of topological gyrogroups is a generalization of a topological group. In this work, ones prove that a topological gyrogroup G is metrizable iff G has an {omega}{omega}-base and G is Frechet-Urysohn. Moreover, in topological gyrogroups, every (countably, sequentially) compact subset being strictly (strongly) Frechet-Urysohn and having an {omega}{omega}-base are all weakly three-space properties with H a closed L-subgyrogroup
A topological group $G$ is said to have a local $omega^omega$-base if the neighbourhood system at identity admits a monotone cofinal map from the directed set $omega^omega$. In particular, every metrizable group is such, but the class of groups with a local $omega^omega$-base is significantly wider. The aim of this article is to better understand the boundaries of this class, by presenting new examples and counter-examples. Ultraproducts and non-arichimedean ordered fields lead to natural families of non-metrizable groups with a local $omega^omega$-base which nevertheless are Baire topological spaces. More examples come from such constructions as the free topological group $F(X)$ and the free Abelian topological group $A(X)$ of a Tychonoff (more generally uniform) space $X$, as well as the free product of topological groups. We show that 1) the free product of countably many separable topological groups with a local $omega^omega$-base admits a local $omega^omega$-base; 2) the group $A(X)$ of a Tychonoff space $X$ admits a local $omega^omega$-base if and only if the finest uniformity of $X$ has a $omega^omega$-base; 3) the group $F(X)$ of a Tychonoff space $X$ admits a local $omega^omega$-base provided $X$ is separable and the finest uniformity of $X$ has a $omega^omega$-base.
In the present paper, we are aiming to study limiting behavior of infinite dimensional Volterra operators. We introduce two classes $tilde {mathcal{V}}^+$ and $tilde{mathcal{V}}^-$of infinite dimensional Volterra operators. For operators taken from the introduced classes we study their omega limiting sets $omega_V$ and $omega_V^{(w)}$ with respect to $ell^1$-norm and pointwise convergence, respectively. To investigate the relations between these limiting sets, we study linear Lyapunov functions for such kind of Volterra operators. It is proven that if Volterra operator belongs to $tilde {mathcal{V}}^+$, then the sets and $omega_V^{(w)}(xb)$ coincide for every $xbin S$, and moreover, they are non empty. If Volterra operator belongs to $tilde {mathcal{V}}^-$, then $omega_V(xb)$ could be empty, and it implies the non-ergodicity (w.r.t $ell^1$-norm) of $V$, while it is weak ergodic.
This note provides a correct proof of the result claimed by the second author that locally compact normal spaces are collectionwise Hausdorff in certain models obtained by forcing with a coherent Souslin tree. A novel feature of the proof is the use of saturation of the non-stationary ideal on omega_1, as well as of a strong form of Changs Conjecture. Together with other improvements, this enables the characterization of locally compact hereditarily paracompact spaces as those locally compact, hereditarily normal spaces that do not include a copy of omega_1.
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